scholarly journals Do we have a sense for irrational numbers?

2017 ◽  
Vol 2 (3) ◽  
pp. 170-189 ◽  
Author(s):  
Andreas Obersteiner ◽  
Veronika Hofreiter
Keyword(s):  
Correct Response ◽  
Response Times ◽  
Number Sense ◽  
Irrational Number ◽  
Rational Numbers ◽  
Cube Root ◽  
Irrational Numbers ◽  
Whole Numbers ◽  
Symbolic Notation ◽  
Whole Number

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

A number pencil

1967 ◽  
Vol 14 (7) ◽  
pp. 557-559
Author(s):  
David M. Clarkson
Keyword(s):  
Elementary Teachers ◽  
Sixth Grade ◽  
Rational Numbers ◽  
Class Discussion ◽  
Irrational Numbers ◽  
Whole Numbers ◽  
Number Lines

So much use is being made of number lines these days that it may not occur to elementary teachers to represent numbers in other ways. There are, in fact, many ways to picture whole numbers geometrically as arrays of squares or triangles or other shapes. Often, important insights into, for example, oddness and evenness can be gained by such representations. The following account of a sixth-grade class discussion of fractions shows how a “number pencil” can be constructed to represent all the positive rational numbers, and, by a similar method, also the negative rationals. An extension of this could even be made to obtain a number pencil picturing certain irrational numbers.

New Publications

1974 ◽  
Vol 67 (2) ◽  
pp. 152-155
Keyword(s):  
Word Problems ◽  
Rational Numbers ◽  
Natural Numbers ◽  
Irrational Numbers ◽  
Whole Numbers

The text provides a refresh on these topics: natural numbers, whole numbers, integers, rational numbers, decimals, and irrational numbers. There are two final chapters on geometry and selected applications. A good many “word” problems are included, along with lots of drill exercises. Exposition is rather brief. The treatment of topics is elementary throughout.— Skeen.

Numbers, numerosities, and new directions

2021 ◽  
Vol 44 ◽  
Author(s):  
Sam Clarke ◽  
Jacob Beck
Keyword(s):  
Number Sense ◽  
Number System ◽  
Rational Numbers ◽  
Theoretical Research ◽  
Approximate Number System ◽  
Irrational Numbers ◽  
Target Article ◽  
New Directions

Abstract In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.

Math Roots: Ratios and Proportions: They Are Not All Greek to Me

2009 ◽  
Vol 14 (6) ◽  
pp. 370-378
Author(s):  
Joanne E. Snow ◽  
Mary K. Porter
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Real Numbers ◽  
Irrational Numbers ◽  
Line Segments ◽  
Whole Numbers ◽  
The Past ◽  
Geometric Objects ◽  
History Of

Today, the concept of number includes the sets of whole numbers, integers, rational numbers, and real numbers. This was not always so. At the time of Euclid (circa 330-270 BC), the only numbers used were whole numbers. To express quantitative relationships among geometric objects, such as line segments, triangles, circles, and spheres, the Greeks used ratios and proportions but not real numbers (fractions or irrational numbers). Although today we have full use of the number system, we still find ratios and proportions useful and effective when comparing quantities. In this article, we examine the history of ratios and proportions and their value to people from the past through the present.

scholarly journals The Number Sense Represents (Rational) Numbers

2021 ◽  
pp. 1-57
Author(s):  
Sam Clarke ◽  
Jacob Beck
Keyword(s):  
Number Sense ◽  
Number System ◽  
Rational Numbers ◽  
Future Research ◽  
Approximate Number System ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Turn On ◽  
Orthodox View

Abstract On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique—the arguments from congruency, confounds, and imprecision—and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

Call for Manuscripts for the 2014 Focus Issue: Rational Number Sense – October 2012

2012 ◽  
Vol 18 (3) ◽  
pp. 189
Keyword(s):  
Rational Number ◽  
Number Sense ◽  
Rational Numbers

This call for manuscripts is requesting articles that address how to make sense of rational numbers in their myriad forms, including as fractions, ratios, rates, percentages, and decimals.

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